Enumerating edge-constrained triangulations and edge-constrained non-crossing geometric spanning trees

In this paper, we present algorithms for enumerating, without repetitions, all triangulations and non-crossing geometric spanning trees on a given set of n points in the plane under edge inclusion constraint (i.e., some edges are required to be included in the graph). We will first extend the lexicographically ordered triangulations introduced by Bespamyatnikh to the edge-constrained case, and then we prove that a set of all edge-constrained non-crossing spanning trees is connected via remove-add flips, based on the edge-constrained lexicographically largest triangulation. More specifically, we prove that all edge-constrained triangulations can be transformed to the lexicographically largest triangulation among them by O(n²) greedy flips, i.e., by greedily increasing the lexicographical ordering of the edge list, and a similar result also holds for a set of edge-constrained non-crossing spanning trees. Our enumeration algorithms generate each output triangulation and non-crossing spanning tree in O(loglogn) and O(n²) time, respectively, based on the reverse search technique.     

Efficient Algorithms for Maximum Regression Depth

We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n ^d ) time and O(n ^d−1) space algorithm computes undirected depth of all points in d dimensions. Properties of undirected depth lead to an O(nlog ² n) time and O(n) space algorithm for computing a point of maximum depth in two dimensions, which has been improved to an O(nlog n) time algorithm by Langerman and Steiger (Discrete Comput. Geom. 30(2):299–309, [2003]). Furthermore, we describe the structure of depth in the plane and higher dimensions, leading to various other geometric and algorithmic results. A preliminary version of this paper appeared in the proceedings of the 15th Annual ACM Symposium on Computational Geometry (1999) M. van Kreveld partially funded by the Netherlands Organization for Scientific Research (NWO) under FOCUS/BRICKS grant number 642.065.503. J.S.B. Mitchell’s research largely conducted while the author was a Fulbright Research Scholar at Tel Aviv University. The author is partially supported by NSF (CCR-9504192, CCR-9732220), Boeing, Bridgeport Machines, Sandia, Seagull Technology, and Sun Microsystems. M. Sharir supported by NSF Grants CCR-97-32101 and CCR-94-24398, by grants from the U.S.–Israeli Binational Science Foundation, the G.I.F., the German–Israeli Foundation for Scientific Research and Development, and the ESPRIT IV LTR project No. 21957 (CGAL), and by the Hermann Minkowski—MINERVA Center for Geometry at Tel Aviv University. J. Snoeyink supported in part by grants from NSERC, the Killam Foundation, and CIES while at the University of British Columbia.     

Optimal Partition Trees

We revisit one of the most fundamental classes of data structure problems in computational geometry: range searching. Matoušek (Discrete Comput. Geom. 10:157–182, 1993) gave a partition tree method for d-dimensional simplex range searching achieving O(n) space and O(n ^1−1/d ) query time. Although this method is generally believed to be optimal, it is complicated and requires O(n ^1+ε ) preprocessing time for any fixed ε>0. An earlier method by Matoušek (Discrete Comput. Geom. 8:315–334, 1992) requires O(nlogn) preprocessing time but O(n ^1−1/d log ^O(1) n) query time. We give a new method that achieves simultaneously O(nlogn) preprocessing time, O(n) space, and O(n ^1−1/d ) query time with high probability. Our method has several advantages:

Almost linear upper bounds on the length of general davenport—schinzel sequences

Davenport—Schinzel sequences are sequences that do not contain forbidden subsequences of alternating symbols. They arise in the computation of the envelope of a set of functions. We obtain almost linear upper bounds on the length λ_s(n) of Davenport—Schinzel sequences composed ofn symbols in which no alternating subsequence is of length greater thans+1. These bounds are of the formO(nα(n)^O(α(n)5-3)), and they generalize and extend the tight bound Θ(nα(n)) obtained by Hart and Sharir for the special cases=3 (α(n) is the functional inverse of Ackermann’s function), and also improve the upper boundO(n log*n) due to Szemerédi. Work on this paper has been supported in part by a grant from the U.S. — Israeli Binational Science Foundation.     

Diameter,width, closest line pair,and parametric searching

We apply Megiddo's parametric searching technique to several geometric optimization problems and derive significantly improved solutions for them. We obtain, for any fixed ε>0, anO(n ^1+ε) algorithm for computing the diameter of a point set in 3-space, anO(^8/5+ε) algorithm for computing the width of such a set, and onO(n ^8/5+ε) algorithm for computing the closest pair in a set ofn lines in space. All these algorithms are deterministic. Work by Bernard Chazelle was supported by NSF Grant CCR-90-02352. Work by Herbert Edelsbrunner was supported by NSF Grant CCR-89-21421. Work by Leonidas Guibas and Micha Sharir was supported by a grant from the U.S.-Israeli Binational Science Foundation. Work by Micha Sharir was also supported by ONR Grant N00014-90-J-1284, by NSF Grant CCR-89-01484, and by grants from the Fund for Basic Research administered by the Israeli Academy of Sciences, and the G.I.F., the German-Israeli Foundation for Scientific Research and Development.     

Improved Bounds on the Union Complexity of Fat Objects

We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ-fat objects. We prove that the union complexity of any set of n such objects is O(λ _s+2(n)log ² n). This improves the best known bound, and extends it to a more general class of objects. The research was supported by the Netherlands’ Organisation for Scientific Research (NWO) under project no. 639.023.301.     

Ordered representations of spaces of integrable functions

Let X be a Banach space, (Ω,Σ) a measurable space and let m : Σ → X be a (countably additive) vector measure. Consider the corresponding space of integrable functions L¹(m). In this paper we analyze the set of (countably additive) vector measures n satisfying that L¹(n) = L¹(m). In order to do this we define a (quasi) order relation on this set to obtain under adequate requirements the simplest representation of the space L¹(m) associated to downward directed subsets of the set of all the representations. This research has been partially supported by La Junta de Andalucía. The support of D.G.I. under project MTM2006–11690–C02 (M.E.C. Spain) and FEDER is gratefully acknowledged.     

Lip Automorphism Germ and Lip Automorphism

Let X be a metric space, ε^n(X) be the standard trivial Lip n-bundle over X, and Φ be a Lip automorphism germ of ε^n(X). This paper proves that there is a Lip automorphism Φ‘ of ε^n(X) such that the germ of Φ‘ is Φ.     

Some basic inequalities in higher dimensional non-Euclid space

In this paper, the concept of a finite mass-points system∑N(H(A))(N>n) being in a sphere in an n-dimensional hyperbolic space Hn and a finite mass-points system∑N(S(A))(N>n) being in a hyperplane in an n-dimensional spherical space Sn is introduced, then, the rank of the Cayley-Menger matrix AN(H)(or a AN(S)) of the finite mass-points system∑∑N(S(A))(or∑N(S(A))) in an n-dimensional hyperbolic space Hn (or spherical space Sn) is no more than n 2 when∑N(H(A))(N>n) (or∑N(S(A))(N>n)) are in a sphere (or hyperplane). On the one hand, the Yang-Zhang's inequalities, the Neuberg-Pedoe's inequalities and the inequality of the metric addition in an n-dimensional hyperbolic space Hn and in an n-dimensional spherical space Sn are established by the method of characteristic roots. These are basic inequalities in hyperbolic geometry and spherical geometry. On the other hand, some relative problems and conjectures are brought.     

Some results on 4<Superscript><Emphasis Type="Italic">m</Emphasis></Superscript>2<Superscript><Emphasis Type="Italic">n</Emphasis></Superscript> designs with clear two-factor interaction components

Clear effects criterion is one of the important rules for selecting optimal fractional factorial designs, and it has become an active research issue in recent years. Tang et al. derived upper and lower bounds on the maximum number of clear two-factor interactions (2fi’s) in 2 ^n−(n−k) fractional factorial designs of resolutions III and IV by constructing a 2 ^n−(n−k) design for given k, which are only restricted for the symmetrical case. This paper proposes and studies the clear effects problem for the asymmetrical case. It improves the construction method of Tang et al. for 2 ^n−(n−k) designs with resolution III and derives the upper and lower bounds on the maximum number of clear two-factor interaction components (2fic’s) in 4 ^m 2 ⁿ designs with resolutions III and IV. The lower bounds are achieved by constructing specific designs. Comparisons show that the number of clear 2fic’s in the resulting design attains its maximum number in many cases, which reveals that the construction methods are satisfactory when they are used to construct 4 ^m 2 ⁿ designs under the clear effects criterion. This work was supported by the National Natural Science Foundation of China (Grant Nos. 10571093, 10671099 and 10771123), the Research Foundation for Doctor Programme (Grant No. 20050055038) and the Natural Science Foundation of Shandong Province of China (Grant No. Q2007A05). Zhang’s research was also supported by the Visiting Scholar Program at Chern Institute of Mathematics.     

On the number of minimal 1-Steiner trees

We count the number of nonisomorphic geometric minimum spanning trees formed by adding a single point to ann-point set ind-dimensional space, by relating it to a family of convex decompositions of space. TheO(n ^d log^{2d 2}−d n) bound that we obtain significantly improves previously known bounds and is tight to within a polylogarithmic factor. The research of D. Eppstein was performed in part while visiting Xerox PARC.     

On the code generated by the incidence matrix of points and hyperplanes in <Emphasis Type="Italic">PG</Emphasis>(<Emphasis Type="Italic">n</Emphasis>,<Emphasis Type="Italic">q</Emphasis>) and its dual

In this paper, we study the p-ary linear code C(PG(n,q)), q = p ^h , p prime, h ≥ 1, generated by the incidence matrix of points and hyperplanes of a Desarguesian projective space PG(n,q), and its dual code. We link the codewords of small weight of this code to blocking sets with respect to lines in PG(n,q) and we exclude all possible codewords arising from small linear blocking sets. We also look at the dual code of C(PG(n,q)) and we prove that finding the minimum weight of the dual code can be reduced to finding the minimum weight of the dual code of points and lines in PG(2,q). We present an improved upper bound on this minimum weight and we show that we can drop the divisibility condition on the weight of the codewords in Sachar’s lower bound (Geom Dedicata 8:407–415, 1979). G. Van de Voorde’s research was supported by the Institute for the Promotion of Innovation through Science and Technology in Flanders (IWT-Vlaanderen).     

On visibility and covering by convex sets

A setX⊆ℝ ^d isn-convex if among anyn of its points there exist two such that the segment connecting them is contained inX. Perles and Shelah have shown that any closed (n+1)-convex set in the plane is the union of at mostn ⁶ convex sets. We improve their bound to 18n ³, and show a lower bound of order Ω(n ²). We also show that ifX⊆ℝ² is ann-convex set such that its complement has λ one-point path-connectivity components, λ<∞, thenX is the union ofO(n ⁴+n ²λ) convex sets. Two other results onn-convex sets are stated in the introduction (Corollary 1.2 and Proposition 1.4). Research supported by Charles University grants GAUK 99/158 and 99/159, and by U.S.-Czechoslovak Science and Technology Program Grant No. 94051. Part of the work by J. Matoušek was done during the author’s visits at Tel Aviv University and The Hebrew University of Jerusalem. Part of the work by P. Valtr was done during his visit at the University of Cambridge supported by EC Network DIMANET/PECO Contract No. ERBCIPDCT 94-0623.     

Analysis of Incomplete Data and an Intrinsic-Dimension Helly Theorem

The analysis of incomplete data is a long-standing challenge in practical statistics. When, as is typical, data objects are represented by points in ℝ ^d , incomplete data objects correspond to affine subspaces (lines or Δ-flats). With this motivation we study the problem of finding the minimum intersection radius r(ℒ) of a set of lines or Δ-flats ℒ: the least r such that there is a ball of radius r intersecting every flat in ℒ. Known algorithms for finding the minimum enclosing ball for a point set (or clustering by several balls) do not easily extend to higher-dimensional flats, primarily because “distances” between flats do not satisfy the triangle inequality. In this paper we show how to restore geometry (i.e., a substitute for the triangle inequality) to the problem, through a new analog of Helly’s theorem. This “intrinsic-dimension” Helly theorem states: for any family ℒ of Δ-dimensional convex sets in a Hilbert space, there exist Δ+2 sets ℒ′⊆ℒ such that r(ℒ)≤2r(ℒ′). Based upon this we present an algorithm that computes a (1+ε)-core set ℒ′⊆ℒ, |ℒ′|=O(Δ ⁴/ε), such that the ball centered at a point c with radius (1+ε)r(ℒ′) intersects every element of ℒ. The running time of the algorithm is O(n ^Δ+1 dpoly (Δ/ε)). For the case of lines or line segments (Δ=1), the (expected) running time of the algorithm can be improved to O(ndpoly (1/ε)). We note that the size of the core set depends only on the dimension of the input objects and is independent of the input size n and the dimension d of the ambient space. An extended abstract appeared in ACM–SIAM Symposium on Discrete Algorithms, 2006. Work was done when J. Gao was with Center for the Mathematics of Information, California Institute of Technology. Work was done when M. Langberg was a postdoctoral scholar at the California Institute of Technology. Research supported in part by NSF grant CCF-0346991. Research of L.J. Schulman supported in part by an NSF ITR and the Okawa Foundation.     

Small Hop-diameter Sparse Spanners for Doubling Metrics

Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a “short” path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse (1+ε)-spanners with small hop diameter for metrics of low doubling dimension. In this paper, we show that given any metric with constant doubling dimension k and any 0<ε<1, one can find (1+ε)-spanner for the metric with nearly linear number of edges (i.e., only O(nlog ^* n+n ε ^−O(k)) edges) and constant hop diameter; we can also obtain a (1+ε)-spanner with linear number of edges (i.e., only n ε ^−O(k) edges) that achieves a hop diameter that grows like the functional inverse of Ackermann’s function. Moreover, we prove that such tradeoffs between the number of edges and the hop diameter are asymptotically optimal. The conference version of the paper appeared in ACM-SIAM SODA 2006. This research of T.-H.H. Chan was done while the author was at Carnegie Mellon University and was partly supported by the NSF grant CCR-0122581 (the ALADDIN project), the NSF CAREER award CCF-0448095, and by an Alfred P. Sloan Fellowship. This research of A. Gupta was partly supported by the NSF grant CCR-0122581 (the ALADDIN project), the NSF CAREER award CCF-0448095, and by an Alfred P. Sloan Fellowship.     

A result on resolutions of Veronese embeddings

This paper deals with syzygies of the ideals of the Veronese embeddings. By Green’s Theorem we know thatO _P ⁿ (d) satisfies Green-Lazarsfeld’s PropertyN _p ∀d≥p, ∀n. By Ottaviani-Paoletti’s theorem ifn≥2, d≥3 and 3d−2≤p thenO _P ⁿ (d) does not satisfy PropertyN _p. The casesn≥3, d≥3, d<p<3d−2 are still open (exceptn=d=3). Here we deal with one of these cases, namely we prove thatO _P ⁿ (3) satisfies PropertyN ₄ ∀n. Besides we prove thatO _P ⁿ (d) satisfiesN _p ∀n≥p iffO _P ⁿ (d) satisfiesN _p.

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Sunto L’argomento di questo articolo sono le sizigie degli ideali delle varietà di Veronese. Per il teorema di Green sappiamo cheO _P ⁿ (d) soddisfa la proprietàN _p di Green-Lazarsfeld ∀d≥p, ∀n. Per il teorema di Ottaviani-Paoletti sen≥2, d≥3 and 3d−2≤p alloraO _P ⁿ (d) non soddisfa la ProprietàN _p. I casin≥3, d≥3, d<p<3d−2 sono ancora aperti (eccetton=d=3). Qui consideriamo uno di tali casi, precisamente proviamo cheO _P ⁿ (3) soddisfa la ProprietàN ₄ ∀n. Inoltre proviamo cheO _P ⁿ (d) soddisfaN _p ∀n≥p se e solo seO _P ^p (d) satisfiesN _p.

     

Efficient binary space partitions for hidden-surface removal and solid modeling

We consider schemes for recursively dividing a set of geometric objects by hyperplanes until all objects are separated. Such abinary space partition, or BSP, is naturally considered as a binary tree where each internal node corresponds to a division. The goal is to choose the hyperplanes properly so that the size of the BSP, i.e., the number of resulting fragments of the objects, is minimized. For the two-dimensional case, we construct BSPs of sizeO(n logn) forn edges, while in three dimensions, we obtain BSPs of sizeO(n ²) forn planar facets and prove a matching lower bound of (n ²). Two applications of efficient BSPs are given. The first is anO(n ²)-sized data structure for implementing a hidden-surface removal scheme of Fuchset al. [6]. The second application is in solid modeling: given a polyhedron described by itsn faces, we show how to generate anO(n ²)-sized CSG (constructive-solid-geometry) formula whose literals correspond to half-spaces supporting the faces of the polyhedron. The best previous results for both of these problems wereO(n ³).This research was done while M. S. Paterson was visiting the Xerox Palo Alto Research Center. This author is supported by a Senior Fellowship of the SERC and by the ESPRIT II BRA Program of the EC under Contract 3075 (ALCOM).     

The weighted maximum independent set problem in permutation graphs

In this paper, sequential and parallel algorithms are presented to find a maximum independent set with largest weight in a weighted permutation graph. The sequential algorithm, which is designed based on dynamic programming, runs in timeO(nlogn) and requiresO(n) space. The parallel algorithm runs inO(log² n) time usingO(n ³/logn) processors on the CREW PRAM, orO(logn) time usingO(n ³) processors on the CRCW PRAM.     

Efficient algorithms for finding minimum spanning trees in undirected and directed graphs

Recently, Fredman and Tarjan invented a new, especially efficient form of heap (priority queue). Their data structure, theFibonacci heap (or F-heap) supports arbitrary deletion inO(logn) amortized time and other heap operations inO(1) amortized time. In this paper we use F-heaps to obtain fast algorithms for finding minimum spanning trees in undirected and directed graphs. For an undirected graph containingn vertices andm edges, our minimum spanning tree algorithm runs inO(m logβ (m, n)) time, improved fromO(mβ(m, n)) time, whereβ(m, n)=min {i|log⁽ⁱ⁾ n ≦m/n}. Our minimum spanning tree algorithm for directed graphs runs inO(n logn + m) time, improved fromO(n log n +m log log log_(m/n+2) n). Both algorithms can be extended to allow a degree constraint at one vertex. Research supported in part by National Science Foundation Grant MCS-8302648. Research supported in part by National Science Foundation Grant MCS-8303139. Research supported in part by National Science Foundation Grant MCS-8300984 and a United States Army Research Office Program Fellowship, DAAG29-83-GO020.     

A Baby Steps/Giant Steps Probabilistic Algorithm for Computing Roadmaps in Smooth Bounded Real Hypersurface

We consider the problem of constructing roadmaps of real algebraic sets. This problem was introduced by Canny to answer connectivity questions and solve motion planning problems. Given s polynomial equations with rational coefficients, of degree D in n variables, Canny’s algorithm has a Monte Carlo cost of sⁿlog(s)D^O(n2)s^{n}\log(s)D^{O(n^{2})} operations in ℚ; a deterministic version runs in time sⁿlog(s)D^O(n4)s^{n}\log(s)D^{O(n^{4})} . A subsequent improvement was due to Basu, Pollack, and Roy, with an algorithm of deterministic cost s^d+1D^O(n2)s^{d+1}D^{O(n^{2})} for the more general problem of computing roadmaps of a semi-algebraic set (d≤n is the dimension of an associated object).